NCERT Solutions Ganita Prakash (Part 1) Chapter 1 Perfect Squares and Odd Numbers — In-text Questions

Book page 6 Updated on2026-09-05

Q1.

Using the pattern above, find 36², given that 35² = 1225.

Answer

1225 is the sum of the first 35 odd numbers, so 36² is that sum plus the 36th odd number.

36th odd number = 2 × 36 – 1 = 71
36² = 1225 + 71 = 1296

Why it happens: Since the sum of the first n odd numbers is n², going from 35² to 36² means adding just one more term of that series. The same step in algebra: (n + 1)² = n² + 2n + 1, and here 2 × 35 + 1 = 71.

Check it yourself: 36 × 36 = 36 × 30 + 36 × 6 = 1080 + 216 = 1296. ✓

Q2.

How do we find the 36th odd number?

Answer

List the odd numbers with their positions and the rule appears at once.

Position n	1	2	3	4	5	6
Odd number	1	3	5	7	9	11

Each odd number is one less than twice its position: 2 × 6 – 1 = 11. So

36th odd number = 2 × 36 – 1 = 71

Why it happens: The even numbers in order are 2, 4, 6, …, so the n-th even number is 2n. Every odd number sits exactly one step below an even number, which gives 2n – 1.

Q3.

What is the nth odd number?

Answer

The n-th odd number is 2n – 1.

n = 1 → 2(1) – 1 = 1
n = 5 → 2(5) – 1 = 9
n = 50 → 2(50) – 1 = 99

Why it happens: Any odd number leaves remainder 1 on division by 2, so it can be written as 2k + 1 for some whole number k ≥ 0. Counting from k = 0 gives 1, 3, 5, …, and the term in position n has k = n – 1, so the value is 2(n – 1) + 1 = 2n – 1.

Tip: This formula is what makes the odd-number test practical. Adding 1 + 3 + 5 + … + (2n – 1) gives n², so the position at which you land on 0 while subtracting is the square root.

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